Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero

Cluckers, Raf; Halupczok, Immanuel

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 45-78.

Par une cascade de généralisations, nous développons une théorie de l’intégration motivique qui fonctionne uniformément dans tous les corps locaux non archimédiens de caractéristique nulle, en surmontant des difficultés reliées à la ramification et à la caractéristique résiduelle petite. Nous définissons une classe de fonctions – appelées fonctions de classe motivique exponentielle – dont nous démontrons qu’elle est stable par intégration et par transformation de Fourier, étendant des résultats et des définitions de [10], [11] et [5]. Nous démontrons des résultats uniformes reliés à la rationalité et à différents types de lieux. Un ingrédient clef est une forme raffinée de l’élimination des quantificateurs de Denef-Pas, qui nous permet de comprendre des ensembles définissables dans le groupe de valeur et dans le corps valué.

Through a cascade of generalizations, we develop a theory of motivic integration which works uniformly in all non-archimedean local fields of characteristic zero, overcoming some of the difficulties related to ramification and small residue field characteristics. We define a class of functions, called functions of motivic exponential class, which we show to be stable under integration and under Fourier transformation, extending results and definitions from [10], [11] and [5]. We prove uniform results related to rationality and to various kinds of loci. A key ingredient is a refined form of Denef-Pas quantifier elimination which allows us to understand definable sets in the value group and in the valued field.

Classification : 14E18, 03C10, 11S80, 11Q25, 40J99
Keywords: Motivic integration, motivic Fourier transforms, motivic exponential functions, $p$-adic integration, non-archimedean geometry, Denef-Pas cell decomposition, quantifier elimination, uniformity in all local fields
Mot clés : Intégration motivique, transformation de Fourier motivique, fonctions motiviques exponentielles, intégration $p$-adique, géométrie non archimédienne, décomposition cellulaire de Denef-Pas, élimination des quantificateurs, uniformité dans tous les corps locaux

[1] Aizenbud, A.; Drinfeld, V. The wave front set of the Fourier transform of algebraic measures, Israel J. Math., Volume 207 (2015) no. 2, pp. 527-580 | MR | Zbl

[2] Basarab, Ş. Relative elimination of quantifiers for Henselian valued fields, Ann. Pure Appl. Logic, Volume 53 (1991) no. 1, pp. 51-74 | DOI | MR | Zbl

[3] Cluckers, R. Presburger sets and $p$-minimal fields, J. Symbolic Logic, Volume 68 (2003), pp. 153-162 | DOI | MR | Zbl

[4] Cluckers, R.; Denef, J. Orbital integrals for linear groups, J. Inst. Math. Jussieu, Volume 7 (2008) no. 2, pp. 269-289 | MR | Zbl

[5] Cluckers, R.; Gordon, J.; Halupczok, I. Integrability of oscillatory functions on local fields: transfer principles, Duke Math. J., Volume 163 (2014) no. 8, pp. 1549-1600 | DOI | MR | Zbl

[6] Cluckers, R.; Gordon, J.; Halupczok, I. Uniform analysis on local fields and applications to orbital integrals (2017) (arXiv:1703.03381)

[7] Cluckers, R.; Lipshitz, L. Fields with analytic structure, J. Eur. Math. Soc. (JEMS), Volume 13 (2011), pp. 1147-1223 | DOI | MR | Zbl

[8] Cluckers, R.; Loeser, F. $b$-minimality, J. Math. Logic, Volume 7 (2007) no. 2, pp. 195-227 | MR | Zbl

[9] Cluckers, R.; Loeser, F. Constructible motivic functions and motivic integration, Invent. Math., Volume 173 (2008) no. 1, pp. 23-121 | MR | Zbl

[10] Cluckers, R.; Loeser, F. Constructible exponential functions, motivic Fourier transform and transfer principle, Ann. of Math. (2), Volume 171 (2010), pp. 1011-1065 | DOI | MR | Zbl

[11] Cluckers, R.; Loeser, F. Motivic integration in all residue field characteristics for Henselian discretely valued fields of characteristic zero, J. reine angew. Math., Volume 701 (2015), pp. 1-31 | MR | Zbl

[12] Cohen, P. J. Decision procedures for real and $p$-adic fields, Comm. Pure Appl. Math., Volume 22 (1969), pp. 131-151 | DOI | MR | Zbl

[13] Denef, J. The rationality of the Poincaré series associated to the $p$-adic points on a variety, Invent. Math., Volume 77 (1984), pp. 1-23 | DOI | Zbl

[14] Denef, J. $p$-adic semialgebraic sets and cell decomposition, J. reine angew. Math., Volume 369 (1986), pp. 154-166

[15] Denef, J. On the Degree of Igusa’s local zeta function, Amer. J. Math., Volume 109 (1987), pp. 991-1008 | DOI | MR | Zbl

[16] Denef, J. Report on Igusa’s local zeta function, Séminaire Bourbaki (Astérisque), Volume 201-203, Société Mathématique de France, 1991, pp. 359-386 | Zbl

[17] Denef, J. Arithmetic and geometric applications of quantifier elimination for valued fields, Model theory, algebra, and geometry (Haskell, D.; Pillay, A.; Steinhorn, C., eds.) (MSRI Publications), Volume 39, Cambridge University Press, 2000, pp. 173-198 | MR

[18] Denef, J.; Loeser, F. Definable sets, motives and $p$-adic integrals, J. Amer. Math. Soc., Volume 14 (2001) no. 2, pp. 429-469 | DOI | MR | Zbl

[19] Flenner, J. Relative decidability and definability in henselian valued fields, J. Symbolic Logic, Volume 76 (2011) no. 4, pp. 1240-1260 | DOI | MR | Zbl

[20] Hironaka, H. Resolution of singularities of an algebraic variety over a field of characteristic zero. I, Ann. of Math. (2), Volume 79 (1964) no. 1, pp. 109-203 | MR | Zbl

[21] Hrushovski, E.; Kazhdan, D. Integration in valued fields, Algebraic geometry and number theory (Progress in Math.), Volume 253, Birkhäuser Boston, Boston, MA, 2006, pp. 261-405 | DOI | MR | Zbl

[22] Igusa, J. An introduction to the theory of local zeta functions, Studies in advanced mathematics, American Mathematical Society, Providence, RI, 2000 | Zbl

[23] Loeser, F.; Sebag, J. Motivic integration on smooth rigid varieties and invariants of degenerations, Duke Math. J., Volume 119 (2003) no. 2, pp. 315-344 | MR | Zbl

[24] Macintyre, A. On definable subsets of $p$-adic fields, J. Symbolic Logic, Volume 41 (1976), pp. 605-610 | DOI | MR | Zbl

[25] Nicaise, J.; Sebag, J. Invariant de Serre et fibre de Milnor analytique, Comptes Rendus Mathématique, Volume 341 (2005) no. 1, pp. 21-24 | DOI | MR | Zbl

[26] Nicaise, J.; Sebag, J. Motivic Serre invariants, ramification, and the analytic Milnor fiber, Invent. Math., Volume 168 (2007) no. 1, pp. 133-173 | MR | Zbl

[27] Pas, J. Uniform $p$-adic cell decomposition and local zeta functions, J. reine angew. Math., Volume 399 (1989), pp. 137-172 | MR | Zbl

[28] Pas, J. Cell decomposition and local zeta functions in a tower of unramified extensions of a $p$-adic field, Proc. London Math. Soc. (3), Volume 60 (1990) no. 1, pp. 37-67 | DOI | MR | Zbl

[29] Rideau, S. Some properties of analytic difference valued fields, J. Inst. Math. Jussieu, Volume 16 (2017) no. 3, pp. 447-499 | DOI | MR | Zbl

[30] Sebag, J. Intégration motivique sur les schémas formels, Bull. Soc. math. France, Volume 132 (2004) no. 1, pp. 1-54 | DOI | Numdam | Zbl